Minsky solidly makes the point that when math is taught as nothing more than an endless series of tedious drills with no purpose in sight, of course it demotivates kids. No one likes pointless Sisyphean tasks.

Many students, however, "rediscover" math when they get to their first class that involves theorems and proofs (often in high-school geometry) and when it starts being used in science coursework. It is then that they realize that math is a way of thinking and this can transform their opinion and motivation for math.

The best teachers find ways to relate mathematics to real life and real purpose. Yeah, it is always going to be challenging, but having a purpose creates motivation to get through the tedium.

I agree with a lot of what you're saying, but not this. The pattern is depressingly clear. People are taught math by teachers who don't really understand it and didn't like it themselves; most of the students wind up in the same place, and so each generation poisons the next. Elementary school teachers especially tend to be drawn from the subset of the population that doesn't like math.

And it starts with the idea that math is all about rote memorization. I would go farther than Marvin did with the flash cards. I would say, for each fact, let's work it out by the standard algorithm, then see if we can find two or three more ways to arrive at it: by distributing multiplication over addition (e.g. 7 × 6 = (6 + 1) × 6 = 6 × 6 + 6 = 36 + 6) or by factoring one multiplicand and reassociating (e.g. 7 × 6 = 7 × 2 × 3 = (7 × 3) × 2 = 21 × 2, which is easy because you can double both digits of 21 without generating a number bigger than 10).

So when I read the reply of the "traditional teacher" to the 6-year-old who had a novel way of computing 15 + 15 -- "Your answer is right but your method was wrong" -- I think that if this was a real anecdote, the teacher should have been fired on the spot, for not understanding the first thing about mathematics! But sadly, no one could probably have been found to replace this teacher who wouldn't have suffered from the same fundamental misconception. It's not about knowing the "right" way: it's about knowing many ways.

Luckily I didn't say that out loud, otherwise they might have made me hate mathematics and I wouldn't love science now (:

Needless to say, my geometry grade was much better than my algebra grade.

Then, in college, taking pre-calc/Trig, it was back to a lot of boring repetition (except trig, that was actually interesting). But then in Calc I, it was fun again. The idea behind derivatives was actually interesting, and hooked me early on in Calc I.

Anyway, I always found that, other than tedium, the biggest challenge I had with math was being put off by cryptic notation. I tend to want to read things to myself in my head as I look at them on a page, and when I'm looking at (new/unfamiliar) math notation, and I don't know how to "pronounce" it to myself, my mind wants to just blank out and skip it.

Material that has an almost unnecessarily comprehensive vocabulary and notation practices guide at the beginning/end is the best kind.

Just look at various cultural traditions, popular (and even serious) music, crafts, sports, ... sneeze ligion. Mindless repetition is everywhere.

The same kid that hates doing arithmetic will play the same video game for hours in which a small variation on the same set of events happens over and over again.

The kids you're describing are the smart minority, of whom a portion will go on into STEM type fields. They too will hit their mathematics ceilings. Most will go as far as a couple of third year undergrad math courses and that's where it ends.

Meanwhile, that dual sense- the pleasure the video game designer tries to create and the sense of importance the priest creates- these are fundamentally absent from our teaching methods in early arithmetic, which, at least in America, we call 'math,' not differentiating the symbolic thought processes from the repetitive arithmetic, incorporating little if any fun, and constantly failing to appropriately create a sense of importance (the most common teenage question in math classes is, "Why do I have to learn this? When am I ever going to use this?").

While there Do exist efforts to instill a video game's sense of fun to arithmetic exercises and a sense of near-religious importance to the symbolic thought of math, we are functionally applying a tourniquet to the mathematical understanding of the young by just force-feeding arithmetic drills and emphasizing mistake avoidance instead of proper thinking- and it's very little wonder that only some few students' mathematical passion survives the tourniquet.

That said, I started over in a community college, and finished up to trig. with ease. I found trigonometry, surprisingly straight forward. I went on to finish a year of physics. I didn't need calculus for my major, and heard it was really hard. Plus, I didn't want to ruin my grade point average with a class I didn't need. I now regret not taking calculus, but it's so much easier to learn something these days.

My struggle with math was two fold. I didn't care in high school, and I never had a firm grasp on basic math. I look back , and once I truly inderstood basic math, especially fractions, and percentages; it all became so easy.

I see a lot of kids still struggling with math. They get pushed along, and avoid any class that has math in it. I do blame our U.S. teaching system in this case. Keep teaching basic math until the kid can teach it to to their classmates.

Then, and only then move on to algebra, and trig? As to calculus, I didn't take it so I won't comment, but when I was applying to health professional schools, I didn't find one that required Calculus. Probally for good reason. My physics classmates biggest, silent stumbling block was they were horrid in math, and looking back--they just didn't know the basics, so every step up the ladder became more mysterious. Most of my classmates seemed to the solving problems in physics by memorizing the homework; not truly understanding the problems.

It's funny to hear you say that. I don't know how old you are, but lately I've been trying to teach myself some additional math that I never took before (like Linear Algebra) and refreshing on Calculus, and now - at the ripe old age of 42 - I swear I find it easier to learn this stuff than I did when I was in college 20 some odd years ago.

Maybe it's just a question of motivation, or maybe I have more context available now, or, hell, maybe I've gotten smarter... but whatever it is, I'm not going to complain, considering how you always hear about how our minds slow down as we age and things are supposed to get harder to learn.

In fact, I've never learnt it. From time to time I still find myself multiplying on fingers; as a kid I used it a lot, now the table went into my head automatically simply from continued usage. The way to multiply two natural numbers from <6; 9> on fingers was one of the most important things my mom tought me :)

There's also a way as presented in the article (87 = 88-8), but I mostly use it when double checking, as I find it slightly more error-prone when doing it in head (and obviously very inefficient when done on paper). Works great for bigger numbers though.

Once in junior high school my math teacher caught me on multiplying on fingers and ordered me to learn the table for the next lession. Managed to pass it, but quickly forgot in following days. I never cared - I was always doing very fast and well on math classes anyway.

Kids are naturally curious and fighting against their curiosity can easily lead to losing their interest.

High school geometry (at least in the US, and in my experience) is the worst of them all! Two-column proofs should be banned.

My personal feeling (as a maths grad student) is that the utility of two column proofs depends on the field. For logic and algorithms sure but estimating integrals it becomes tedious quickly.

[1] http://research.microsoft.com/en-us/um/people/lamport/pubs/p...

Geometry is taught correctly when proofs are used as a tool to convey a deep intuition about a mathematical pattern. Check out Lockhart's book [1] if you want to see what it's like when done right, though there are many more examples. Two-column proofs are simply a tool for the lazy/unknowledgeable teachers to fill a geometry class.

I am aware of Lamport's work, and it's a specific tool for a specific subfield in which there is a plethora of false results. Ignoring the fact that most of theoretical computer science research and most of math research more does not fall into the category that Lamport is critical of (distributed computing), these temporary issues about rigor in academic publishing should have no effect on high school pedagogy. Instead, we should listen to the world's finest math teachers, who pretty much all agree that two-column proofs are awful. A quote from Lockhart [2]:

> Geometry class is by far the most mentally and emotionally destructive component of the entire K-12 mathematics curriculum. Other math courses may hide the beautiful bird, or put it in a cage, but in geometry class it is openly and cruelly tortured. What is happening is the systematic undermining of the student’s intuition. A proof, that is, a mathematical argument, is a work of fiction, a poem. Its goal is to satisfy. A beautiful proof should explain, and it should explain clearly, deeply, and elegantly. A well-written, well-crafted argument should feel like a splash of cool water, and be a beacon of light— it should refresh the spirit and illuminate the mind. And it should be charming. There is nothing charming about what passes for proof in geometry class. Students are presented a rigid and dogmatic format in which their so-called “proofs” are to be conducted— a format as unnecessary and inappropriate as insisting that children who wish to plant a garden refer to their flowers by genus and species.

[1]: http://www.amazon.com/Measurement-Paul-Lockhart/dp/067428438... [2]: https://www.maa.org/external_archive/devlin/LockhartsLament....

Unfortunately, most high school math teachers are undertrained, underpaid, overworked, and pressured to focus on exams. The two-column proof is a crutch. It's not the only awful part of high school math education, nor is it the sole bane of a student's math education (I never said it was). But it is the most egregious example of bad math education.

I had never heard of that monstrosity before and having googled it, it looks worse than programming in COBOL.

Also: it runs on machines that look like Cylons. You can't beat that. :p

To contrast, LOCAL-STORAGE seems like a regular global variable (.DATA/.BSS) segment. So is WORKING-STORAGE basically like a set of seeming global variables, that actually point into (the equivalent of) a shared mmap'ed file, such that other processes in the same session/pgroup end up mmap'ing the same file and thus sharing memory that way, without having to clone around the memory handle itself via forking/threading/IPC?

In this case, it sort of reminds me of the OS-level version of an anonymous Erlang ETS table held by a supervisor and passed to all the worker children to manipulate.

COBOL's type definitions are also patterns really - for instance you declare a field to be of type 9(7) which means a numeric field of length 7 and so on.

The whole setup gives you very fine control over the structure of your data. For example, you can describe lines of a document to be printed out with strongly typed fields to be filled in by your program, so a form basically.

It's also used as a template for documents like VSAM files. Those are flat-file databases and the COBOL program's working storage imposes structure on them.

Here's a small example:

01 FILE-RECORD.

__03 WS-ACCOUNT_____PIC 9(10).

__03 WS-SEPARATOR1__PIC X.

__03 WS-NAME________PIC S(25).

__03 WS-SEPARATOR2__PIC X.

[Underscores are formatting only, not COBOL syntax]

That declares a structure called FILE-RECORD with four named fields, one of numeric type with length 10, one of alphabetic type with length 25 and two of alphanumeric type with length 1. So that's one record in your file structure and you can fill it in with data from VSAM files and the like, then print it out, with the separator fields for formatting.

It's a strange thing, low-level and high-level at the same time, in a very nice way. It kind of grows on you. Well, it's grown on me anyway. Of course, YMMMV.

Btw, the numbers (01, 03) make the hierarchical relation between the root of the data structure and each of its fields explicit; I won't say I'm in love with that sort of syntax :)

Also, I'm a total no0b still, so I might have made some mistake above, apologies in advance if that is the case.

I had those in high school and I hardly see any problem with them. It's just writing the proof neatly organized into a table, with the reasons for each step given.

High school geometry introduces many students to the idea that math is a way of reasoning and this is _very_ different from the boring grind that they were used to.

And, on the converse, in previous centuries, you'd often see people only start learning math from the beginning in their late teens or early twenties—the time when they first attended "university" without previously attending a grammar or seminary school. These 20-year-olds would be able to pick math all up quite quickly, proceeding from addition and multiplication to calculus and beyond within a year or two.

This suggests to me that what we think of as "mathematical aptitude" is far more a measure of developmental age than anything else. I think a large part of why kids hate math is that we try to cram concepts into their brains when those brains aren't yet the right "shape" to take in those concepts. I expect that offsetting every part of the mathematical curriculum at least two years upward—if not far more—would do wonders for attrition rates.

My belief is that we could even fit all the same curriculum in by the end of public high school: although you'd be "squishing" more of the learning into the later years, the stronger minds kids would have at a higher age would allow them to acquire each successive concept both more quickly and more thoroughly, serving as a better base for the learning going forward, which would in turn be accelerated by that base.

[1] http://www.amazon.com/Knowing-Teaching-Elementary-Mathematic...

http://condor.depaul.edu/sepp/mat660/Askey.pdf

http://www.ams.org/notices/199908/rev-howe.pdf

If you've ever tried to tutor someone who "struggles with word-problems", the majority of the time it's because they don't actually understand the things they've been "learning" at all.

Similarly, this tends to be the difference between people who "get" programming and people who don't. If you ever talked to someone who failed a programming course, it's usually clear from their approach that every new language or API they learned became, to them, another set of rote facts to burn into their mind, never coming together to paint a picture of what the formal act of "programming" is generally.

Instead of memorizing multiplication tables, you instead have to memorize derivative formulas, or the formulas for the Fourier and Laplace transforms, or the difference between a bijection and an injection, or all the conditions on an elliptic curve that make it suitable for cryptography. I could go on and on and on.

The stupid notations that mathematicians use doesn't help either. All math should be done using S expressions (and I'm pretty sure Minsky would have agreed!)

R.I.P. Marvin.

And BTW, the notation actually sucks for paper-and-pencil too because of its ambiguity. See:

http://mitpress.mit.edu/sites/default/files/titles/content/s...

I'm sure that isn't in R5RS and I don't see it in R7RS. I found a description in the Racket manual.

In any case, I like how they are making the context explicit. In the regular math notation for the physics, you just say something like "the Lagrangian for the free particle" and then just spew out something with free variables sticking out of it, like velocity "v". The hacker in me of course asks, how the heck does that work if I have two particles? There is only one v? When we have two, we hand-wave in some subscripts: v1, v2, ... This system makes it explicit: it's the Lagrangian for a particle. So the particle appears as a parameter, and that parameter has accessors on it to retrieve its properties: velocity of the particle, etc.

There's no correlation at all between being able to add up a grocery bill in your head and being good at abstraction.

My accountant is very good at mental arithmetic. I don't expect him to win the Fields Medal any time soon.

This is more relevant than it should be. Too many people leave school thinking math is mental arithmetic, and have no idea what symbolic manipulation is, or why it's useful.

The next step is algebra at 8th or 9th grade. Letters and numbers confuses people too.

In my case I succumbed at Analysis. I like applied math, but dislike constructing proofs.

In high school it's not so bad -- only 28% of high school math didn't major in math, and only 12% of our total math teachers in high school have no qualifications whatsoever. [1] In middle school 55% of math teachers didn't major in math, and about 30% didn't major, minor, or get a certificate in math! [2]

So -- you're right.

[1] https://nces.ed.gov/fastfacts/display.asp?id=58

[2] http://nces.ed.gov/pubs2015/2015815.pdf